The longissimus is a muscle, or set of muscles, that runs parallel to the spine (referring to the diagram at that Wikipedia link may be useful.) The longissimus dorsi is a particular muscle (or pair of muscles, with one on each side) and the "complex" part implies that there are several adjacent muscles that tend to work together rather than just it in isolation.
This theme of muscles all over the body working together rather than in isolation is common in Rippetoe's books.
To provide further context, the longissimus dorsi of the cow is most often sold by a butcher as the striploin.